Geometry (from the Ancient Greek: γεωμετρία; geo- "earth", -metron "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is called a geometer.
Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts.During the 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is Gauss' Theorema Egregium ("remarkable theorem") that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in an Euclidean space. This implies that surfaces can be studied intrinsically, that is as stand alone spaces, and has been expanded into the theory of manifolds and Riemannian geometry.
Later in the 19th century, it appeared that geometries without the parallel postulate (non-Euclidean geometries) can be developed without introducing any contradiction. The geometry that underlies general relativity is a famous application of non-Euclidean geometry.
Since then, the scope of geometry has been greatly expanded, and the field has been split in many subfields that depend on the underlying methods—differential geometry, algebraic geometry, computational geometry, algebraic topology, discrete geometry (also known as combinatorial geometry), etc.—or on the properties of Euclidean spaces that are disregarded—projective geometry that consider only alignment of points but not distance and parallelism, affine geometry that omits the concept of angle and distance, finite geometry that omits continuity, etc.
Originally developed to model the physical world, geometry has applications in almost all sciences, and also in art, architecture, and other activities that are related to graphics. Geometry also has applications in areas of mathematics that are apparently unrelated. For example, methods of algebraic geometry are fundamental in Wiles's proof of Fermat's Last Theorem, a problem that was stated in terms of elementary arithmetic, and remained unsolved for several centuries.
Homework Statement
ABC is a triangle with I as the centre of the incircle and K as the centre of the circumcircle (I ≠ K). d(X) is the sum of the distances from a point X inside the triangle to the three sides. Verify if d(P)=d(Q) (P and Q are points inside the triangle), the lines PQ and IK...
Homework Statement
http://i.imgur.com/lnk7e0D.png CDE is an equilateral triangle inside a circle, with side length 16. FGH is also an equilateral triangle and F is the mid point of DE. Find the length of the side FGH. Should be expressed as Asqrt(B)-C. Where A B and C are positive integers...
Homework Statement
http://i.imgur.com/lzTN7If.png
Excuse the bad drawing
Point E lies on the side AC of the square ACBD. The segment EB is broken up into 4 equal parts as well as the segment ED. If JK = 3 find the area of the quadrilateral FGKJ[/B]
Homework Equations
Trapezoid equation to...
Hello fellow science enthusiasts! I will start by describing my problem. I am majoring in Computer Science. More importantly, I am taking a course in Single Variable Calculus I. We are using the textbook written by Stewart: Calculus Early Transcendentals. And as I continue read through it...
Hi all,
I'm interested in the behavior of electric fields in a gravitational shockwave geometry. I'm specifically thinking about gravitational shockwaves due to null shells as discussed, for example, in Dray-'tHooft http://www.sciencedirect.com/science/article/pii/0550321385905255 (available...
Homework Statement
Basically there are 3 plate capacitors, with a distance a between the first two and b between the last two. In between the first two plates is voltage V. A particle of charge q is released from the first plate, and when it leaves the second plate it is no longer...
I'm planning to write about analytical geometry, but I'm unsure about where to start. The course that I've taken begins with the definition of a vector, it then proceeds to develops all the vector algebra and all properties. After it reaches the cross product and triple product it begins to...
I think this will probably turn out to be one of the most important QG papers this quarter. It is based on some very interesting and groundbreaking work by Dittrich and Geiller that has come out over the past year and a half. I'll give links to that after posting the main abstract...
"Then, since on the circumference of each of the circles ABDC and ACK two points A and C have been taken at random, the straight line joining the points falls within each circle, but it fell within the circle ABCD and outside ACK, which is absurd. Therefore a circle does not touch a circle...
I don't fully understand hyperbolic space. I saw a numberphile video about it. I thought It would a cool idea to make a video game based around hyperbolic space. I was going make it in html5/css/javascript. I know I need to learn a lot of math.
I was going to render the hyperbolic objects in a...
Hi, Just wondering. Has anyone tried Harold Jacobs Geometry first edition (1974). I've considered getting it instead of 2nd edition, but don't know how big the difference is between them.
In a trivial optimization problem, when seeking the value of x2 that minimizes y(x2)/(x2-x1), the solution is graphically given by the tangent line shown in the figure.
I'm having a lot of difficulty understanding why this is true, i.e., the logical steps behind the equivalence supporting the...
Say you have a box with a round marble on it. Classically I would have no problem describing that, but how is it done in general relativity? Do one just assume that the marble is round and that the box is ... boxy... in the 3D subspace at a certain foil of time? (Foliation is the correct term...
Hi I'm looking for 2 books
1. a big geometry and trigonometry book that covers almost everything (also proofs) from basic to intermediate so i have a solid understanding of geometry trigonometry.
2. a geometry or trigonometry that gives you an appreciation for trigonometry fx how it was used...
Hello, I am studying for an analytic geometry final but I am totally lost for this problem... We didn't even cover this topic in class (my prof didn't show up for class for two weeks) and I have no clue on how to do it. If anyone could help I would appreciate it.
Question: Prove that the...
Homework Statement
the radius from the symmetry to center of the plasma is about 6.2 metres and the minor radius is 2 metres
Homework Equations
Can you guys help me to make the plasma geometry for MCNP?
The Attempt at a Solution
the softcode of plasma geometry
What is the relationship between the nitrogen inversion (or "flip-flop" or "turning itself inside out") and the associated microwave radiation of the ammonia molecule ?
Homework Statement
Let g_{\mu\nu} be a static metric, \partial_t g_{\mu\nu}=0 where t is coordinate time. Show that the metric induced on a spacelike hypersurface t=\textrm{const} is given by
\gamma_{ij} = g_{ij} - \frac{g_{ti} g_{tj}}{g_{tt}} .
Homework Equations
Let y^i be the coordinates...
In the euclidean plane, point A is on a circle centered at point O and point O is on a circle centered at point A. What is measure of angel BAC?
So I drew a picture, and it seems that BAC is going to definitely be greater than 90 degree's. From there I am confused on what to do next. Anyone...
What is the greatest possible area of a triangle that has one vertex at the center of a circle with radius 1 and the other 2 vertices on the circumference of the circle? Can anyone give me some insight into this?
I was thinking maybe make the center angel 90 degree's? If so my circle would have...
i found that in Anechoic chamber there are pyramidal structures over the walls,
what are the geometries of the height and base of those pyramids?
W and H on the picture:
I had vaguely read somewhere a while ago that Ed Witten was developing some sort of new geometry where the only true, fundamental objects were worldsheets. Perhaps I'm not correctly remembering the idea but I think it was something like this. Has he announced this recently? I can't seem to...
I've been quite under desperate lately. I got no clue on how to solve this (apparently easy) geometry problem:
Consider a circle C of center O in a plane and a line segment AB not necessarily outside the circle C.
Having no compass, just a pen and a simple ruler, construct 2 points M and N...
My understanding:
When we draw a triangle on a flat piece of paper and measure the angles using a protractor, the sum of the angles is ##180^\circ##. So we conclude that the universe is locally flat. Suppose we draw a very big triangle that spans across galaxies (say, using lasers and mirrors)...
Homework Statement
See attached picture
Homework Equations
For any rectangular prism, the formulas are the following:
Surface Area = 2(lw + wh + hl); l is length, w is width, and h is height.
Volume = lwh; l is length, w is width, and h is height.
For a square-based prism, the formulas are...
Homework Statement
I need to find optical power (reciprocal focal length) of this system with thin lens
Homework Equations
I tried to solve this using spherical diopter equation
n1/a+n2/b=(n2-n1)/R
where a is object distance and b is image distance
The Attempt at a Solution
equation for...
I'm performing a 3D CFD simulation in ansys fluent on a laboratory in order to verify the pressure map and overall ventilation design. During meshing, ansys is displaying the following error "The mesh file exporter does not support overlapping geometry in Contact Regions. Please resolve the...
In my textbook, its given that the equation of family of circles touching a given circle S and line L is ##S+\lambda L=0##
So to find the equation of family of circles touching line L at point P(p,q), can i use the same equation taking S to be a circle of radius zero and center at P?
That is...
Homework Statement
a) Suppose that A,B,C,D are four "points" in a projective plane, no three of which are on a "line." Consider the "lines" AB, BC, CD, DA. Show that if AB and BC have a common point E, then E = B.
b) From a) deduce that the three lines AB, BC, CD have no common point , and the...
Maybe a little bit of a fresh mathematical virgin thing to ask, but what exactly is the definition of analytical geometry?
I am asking for the reason I have recently acquired a book on the subject, but I found it by accident, and I would like to comprehend the most basic concepts of it!
I am...
Atyy spotted 3 potentially important LQG papers and added them to the biblio thread. The B-O approximation has proven essential in quantum chemistry and similar applications dealing with wave functions of complex systems with many components. Geometry is such a system and it seems reasonable...
Hi everyone,
I have an undergrad in Physics from an Indian University. And presently I am pursuing a masters in theoretical physics in UK.
Here I got a chance to explore courses in the math department. I took two courses in geometry : Differential geometry and Geometry of General Relativity. I...
Homework Statement
The corners A, B and C of a triangle lies on a circle with radius 3. We say the triangle is inscribed in the circle. ∠A is 40° and ∠B is 80°.
Find the length of the sides AB, BC and AC.
Homework EquationsThe Attempt at a Solution
I found out the arc AB is 2π, arc BC is 4π/3...
I am currently looking at grad schools, and I am wondering if anyone knew who are the leading researchers in differential geometry. I know that question is a little vague considering how diverse differential geometry is, but I was hoping that something could direct me in the right direction...
Homework Statement
Picture: http://matematikk.net/res/eksamen/1T/kort/1T_V11.pdf
Task 5, the one the with a triangle inside a square. I'ts not in English so i'll transalate. I managed to do task a
The picture above shows a square ABCD. The sides in the square have length 1. E is the center of...
Homework Statement
One of the sides of a triangle is 7.0cm, another side is 11.0cm.
A Decide the biggest area this triangle can have.
B Make calculations and show how the triangle could look like if the area is 30 square cm.
Homework Equations
Area of a triangle: 0.5*g*h or 0.5*a*b*sinV
The...
Hello all, I was accepted into a program that allows me to take a free college course and I was hoping to take a math/science course. I am most interested in Physics, but I don't know Calc so that isn't really an option, and I was thinking of taking Calculus and Analytic Geometry. However, I am...
Hello I do not fully grasp the concept of Riemann geometry.Please can you use mathematical descriptions but explain them because I am only a 10 year old but of course I know significantly the theories of dimensions. Thank you.
Hi,
I was just wondering if someone could help clarify how pi - theta = phi?
That is the link to the youtube video I was watching, the guys pretty good check him out if you want to learn how to derive the differential scattering cross section.
I'm quite aware of how to compute how FAR you are from the horizon, but my question is, how WIDE is the observable horizon at sea level (like, from left to right, how many kilometers is this):
http://www.jeicentral.com/wp-content/uploads/2014/10/far_sunset_in_te_ocean_horizon-wide.jpg
Thanks!
Hello everyone
This is sort of a geometry problem. I'm sure it has an easy answer but it just won't come to me. Here's my problem
A close packed colloidal aggregate of smaller spherical colloidal particles can be thought of as small spheres within a sphere.
I have the relationship...
Given a cyclic quadrilateral $PQRS$ where $PQ=p,\,QR=q,\,RS=r$, $\angle PQR=120^{\circ}$ and $\angle PQS=30^{\circ}$.
Prove that $|\sqrt{r+p}-\sqrt{r+q}|=\sqrt{r-p-q}$
The title above give my name. I am a pure maths PhD with an interest in physics and geometry. I am currently studying physics for fun and I am very interested in current progress.
I am especially interested in quantisation of space time, holographic theories and dualities.
Regards
John
Homework Statement
What is the largest possible radius of a sphere which is inscribed in a regular tetrahedron
a=10 ( this is the side of the tetrahedron)
r=?
r=5*√6/6
Homework EquationsThe Attempt at a Solution
So first I calculated the Height of pyramid
a2=(2/3*va)2+h2
h=√(a2-(2/3*a*√3/2)2)...
Hello!
We are working with some ferrite rods to build solenoid magnetorquers for a cubesat design. The geometry on these rods is unique. Are there benefits to this type of a geometry?
They are 19.6 cm length x .67cm diameter.
Homework Statement
[/B]
A parallel quadratic slab of glass (n = 1.55 and thickness d = 2 cm, L = 21 cm) rests on a large slab of glass (n = 1.55). To prevent the optical contact weld forming between the two polished surfaces, a small teflon ball (D = 1 cm) is inserted between the slabs on one...
< Mentor Note -- thread moved to HH from the technical math forums, so no HH Template is shown >
Find the equation of a line passing through point A (1, 2) and whose perpendicular distance from origin is maximum.
Homework Statement
In the attached drawing, find R in terms of L and c. Also, at the bottom of the picture I wrote something wrong. I said c, which equals 0.5, is the arc-length of each semi-circle, but I really meant to say each quarter circle. My bad. I'm not given a number for L so that can...